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A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

%I A258593 #14 Feb 16 2025 08:33:25
%S A258593 1,8,46,208,805,2776,8742,25584,70450,184232,460832,1108848,2578295,
%T A258593 5814992,12760598,27317056,57174768,117223008,235818894,466154416,
%U A258593 906606234,1736736024,3280271526,6114139616,11255369609,20478505104,36849912318,65619691088
%N A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%C A258593 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A258593 Vaclav Kotesovec, <a href="/A258593/b258593.txt">Table of n, a(n) for n = 0..10000</a>
%H A258593 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A258593 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A258593 Expansion of (f(x^2)^2 / (chi(-x^2) * phi(-x)^2))^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
%F A258593 Expansion of q^(-1/2) * (eta(q^4)^7 / (eta(q)^4 * eta(q^2) * eta(q^8)^2))^2 in powers of q.
%F A258593 Euler transform of period 8 sequence [ 8, 10, 8, -4, 8, 10, 8, 0, ...].
%F A258593 -4 * a(n) = A260186(2*n + 1).
%F A258593 a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - _Vaclav Kotesovec_, Nov 15 2017
%e A258593 G.f. = 1 + 8*x + 46*x^2 + 208*x^3 + 805*x^4 + 2776*x^5 + 8742*x^6 + ...
%e A258593 G.f. = q + 8*q^3 + 46*q^5 + 208*q^7 + 805*q^9 + 2776*q^11 + 8742*q^13 + ...
%t A258593 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) (EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0 ,x]^2)^2, {x, 0, n}];
%t A258593 a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^2]^2 QPochhammer[ -x^2, x^2] / EllipticTheta[ 4, 0, x]^2)^2, {x, 0, n}];
%o A258593 (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^7 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^8 + A)^2))^2, n))};
%Y A258593 Cf. A260186.
%K A258593 nonn
%O A258593 0,2
%A A258593 _Michael Somos_, Nov 06 2015